复合负二项风险模型的破产概率

成都理工大学硕士学位论文复合负二项风险模型的破产概率姓名：董秀丽申请学位级别：硕士专业：计算数学指导教师：魏贵民;范安东20090501摘要I复合负二项风险模型的破产概率复合负二项风险模型的破产概率复合负二项风险模型的破产概率复合负二项风险模型的破产概率作者:董秀丽女1980年8月计算数学专业师从魏贵民教授,2009年6月毕业于成都理工大学信息管理学院，理学硕士学位。摘摘摘摘要要要要风险理论是现代精算和数学界研究的热点，破产理论是风险理论的核心内容.破产理论的研究溯源于瑞典精算师FlipLundberg于1903年发表的博士论文，他首次在这篇论文中提出了一类最重要的随机过程--Poisson过程.Cramer将Lundberg的工作建立在坚实的数学基础上，形成了经典风险理论的基本理论.如今，对经典模型已有了很透彻的研究，得出了很多重要的结果，为破产理论的发展奠定了坚实的基础，但是由于它本身还存在许多缺陷，所以很多学者对其进行了推广.本文首先对风险理论的研究和发展进行了概述，经典风险模型作为风险理论的核心内容，对其进行了详尽的说明，并综述了风险模型在国内外的研究现状，及经典离散风险模型的组成部分和主要结果.针对目前保险业务逐渐复杂和细化的实际情况，提出了复合负二项风险模型，研究了此模型的破产参数，以期能够更真实更准确的反映保险公司的实际运营情况，便于保险公司做出统筹安排.在保险业务各种复杂的问题中，保险人依照风险的某些特征对其进行分类，但是被划入同一类中的保单仍然不可避免的存在着某种程度的非同质性.因此，对于同一类保单组合的索赔次数模型的描述，首先要确定保单组合中各个保单的索赔次数模型，然后根据同一类保单中的非同质性，确定其模型中的参数分布规律，最后再完整地描述该保单组合的索赔次数模型.本文首先介绍了复合负二项风险模型的生存概率、最终破产概率、有限时间内的生存概率;接着将复合负二项风险模型中保费收取次数及其每次收取的保费都推广为随机变量，提出了单险种复合双负二项风险模型，研究了该模型中盈余过程的数字特征、有限时间内的破产概率、最终破产概率等问题.鉴于目前保险实务中保险险种的逐渐增多，将单险种复合双负二项风险模型进一步推广，提出了双险种的复合双负二项风险模型，研究了该模型中盈利过程的数字特征、最终破产概率、Lundberg不等式等问题.事实上，保险公司的运营过程中会时不时地出现一些随机因素，使得保险公司有些不确定的收益或者支出，为此，研究了带投资和干扰的双险种复合双负二项风险模型的破产问题.在上述每种模型下，都得到了相应的盈余序列的性质，即盈余过程是一个平稳的独立增量过程，得到成都理工大学硕士学位论文II了相应盈余序列的数字特征，利用递推方法和鞅的方法对各模型进行了比较全面的研究，并得到了与经典风险模型极为相似的结论。破产概率的表达式:()()()expRueuERUTTψ-=-∞尤其重要的是找到了破产概率的上界，即Lundberg不等式()Ruueψ-≤其较强的可操作性在保险系统的风险分析中被广泛应用，具有重要的理论和实际意义!关键词:风险模型负二项分布鞅论停时破产概率生存概率盈余投资干扰项AbsractIIITheruinprobabityoncompoundNegativeBinomialRiskModelIntroductionoftheauthor:Dongxiuli,female,wasborninAugust,1980whosetutorwasProfessorWeiGuimin.ShegraduatedfromChengduUniversityofTechnologyinComputingMathematicsmajorandwasgrantedtheMasterDegreeinJune,2009.AbstractRisktheoryisahotstudytopicintherealmofmodernactuarialscienceandmathematics,andruintheoryisthecoreofrisktheory.ThestudyoftherisktheorystartedfromthePhDpaperofFlipLundberg,whowasaSwedishactuarialscientist,in1903,hefirstbroughtupanimportantstochasticprocess,Poissonprocess,inhispaper.CramerdevelopedLundberg'workbasedonthemathematics,andformedthebasetheoryoftheclassicalrisktheory.Atpresent,theclassicalriskmodelhasbeendeeplyresearchedandmanyimportanttheorieshavebeeneduced,whichhaveestablishedastablefoundationforthefurtherdevelopmentoftherisktheory.Asthetheoryhasmanydefects,manystudiershavebeenimprovingit.Thisarticlehasoutlinedtheresearchanddevelopmentoftherisktheory,andsummarizedthenegativebinomialriskmodelindetailinthedomesticandoverseaspresentresearchsituationaswellastheconstituentandthemainresultsoftheclassicsseparateriskmodel.Basedonthis,inviewofthegraduallycomplexandspecificsituationonpresentactualinsurancebusiness,inthisarticleIhaveproposedthemixednegativebinomialriskmodel,andstudiedthebankruptparameterofthismodel,sothattheactualoperationsituationofinsurancecompanycanbereflectedmoretrulyandaccurately,andit'shelpfulfortheinsurancecompanytomaketheoverallplanarrangement.Inallkindsofcomplicatedissuesoftheinsurancebusiness,theinsurercarriesontheclassificationaccordingtotheriskcertaincharacteristics,butthechitinthesamekindstillisinevitabletoexistsomekindofdegreethenon-homogeneity.Therefore,regardingtheclaimnumberoftimesmodel'sdescriptionoftheindenticalkindofchitcombination,firstlymustdeterminatetheclaimnumberoftimesmodelofeachchitinthedefinitepolicycombination,theninthebasisidenticalkindofchit'snon-homogeneity,determinatetheparameterdistributionruleinitsmodel,finallydescribetheclaimnumberoftimesmodelofthischitcombinationcompletelyagain.Thisarticlefirstlyintroducedthesurvivalprobability、thefinalruinprobability成都理工大学硕士学位论文IVofthecompoundnegativebinomialriskmodel,thesurvivalprobabilityinlimitedtime;thenmadetheinsurancepremiumcollectiontimesandeachtimegathersinthenegativebinomialriskmodelberandomvariable,proposedthecompounddoublenegativebinomialriskmodel,studiedthedigitalcharacteristicsoftheearningsprocessinthismodel,theruinprobabilityinthelimitedtime,finalruinprobabilityandsoon.Inviewofthegradualincreasesoftheinsurancetypesinthepresentinsurancepractice,Ipromotedthecompounddoublenegativebinomialriskmodelfurther,proposedthecompoundmufti-negativebinomialriskmodelofdoubleinsurancetypes,andstudiedthedigitalcharacteristicsofearningprocessinthismodel,finalruinprobability,Lundberginequalityandsoon.Infact,intheprocessoftheinsurancecompany'soperation,therearesomerandomfactorsonceforawhile,whichmakesinsurancecompanyhavesomeuncertainincomesorthedisbursements,soIalsostudiedthebankruptproblemsofthedoublecompoundnegativebinomialriskmodelindoubleinsurancetypeswithdisturbanceitems.Undertheaboveeachkindofmodel,allareobtainedthecorrespondingearningssequencenature,namelytheearningsprocessisasteadyindependentincrementprocess,whichobtainedtheconcreteexpressionformofruinprobabilitybyrecursivemethodandMartingalemethod()()()expRueuERUTTψ-=-∞Especiallymoreimportantlyhadfoundtheupperboundaryofruinprobability,thatistheLundberginequality()Ruueψ-≤Itiswidelyappliedintheinsurancesystem'sriskanalysisforitsstrongfeasibility,whichhastheimportanttheoriticalandthepracticalsignificance!Keywords:RiskmodelNegativebinomialdistributionBankruptcyprobabilitymartingalestoppingtimeSurvivalp